Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems

• Published in: Computing Vol. 92; no. 1; pp. 1 - 32
• Main Authors: Mukherjee, Kaushik, Natesan, Srinivasan
• Format: Journal Article
• Language: English
• Published: Vienna Springer Vienna 01.05.2011; Springer; Springer Nature B.V
• Subjects: Acceleration of convergence; Accuracy; Algorithmics. Computability. Computer arithmetics; Applied sciences; Artificial Intelligence; Boundary layer; Computer Appl. in Administrative Data Processing; Computer Communication Networks; Computer Science; Computer science; control theory; systems; Diffusion; Exact sciences and technology; Information Systems Applications (incl.Internet); Mathematical analysis; Mathematics; Methods of scientific computing (including symbolic computation, algebraic computation); Numerical analysis; Numerical analysis. Scientific computation; Ordinary differential equations; Parabolas; Reynolds number; Sciences and techniques of general use; Software Engineering; Studies; Theoretical computing; Variables; Uniform convergence; Piecewise-uniform Shishkin mesh; 65M06; CR G1.8; Richardson extrapolation; Singularly perturbed parabolic problem; Regular boundary layer; 65M12; Upwind scheme; Differential equation; Convection diffusion equation; Euler scheme; Partial differential equation; Parabolic equation; Boundary layer; Grid pattern; Convergence acceleration; Runge Kutta method; Algorithm; Discretization method; Extrapolation; Numerical analysis; One-dimensional calculations; Scientific computation; Multistep method
• ISSN: 0010-485X; 1436-5057
• DOI: 10.1007/s00607-010-0126-8

This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection–diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the ε -uniform accuracy of simple upwinding in the discrete supremum norm from O ( N −1 ln N + Δ t ) to O ( N −2 ln 2 N + Δ t 2 ), where N is the number of mesh-intervals in the spatial direction and Δ t is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.